{
  "id": "1808.05969",
  "version": "v1",
  "published": "2018-08-17T18:52:09.000Z",
  "updated": "2018-08-17T18:52:09.000Z",
  "title": "Stationary points in coalescing stochastic flows on $\\mathbb{R}$",
  "authors": [
    "Andrey A. Dorogovtsev",
    "Georgii V. Riabov",
    "Björn Schmalfuß"
  ],
  "comment": "16 pages, 2 figures",
  "categories": [
    "math.PR",
    "math.DS"
  ],
  "abstract": "This work is devoted to long-time properties of the Arratia flow with drift -- a stochastic flow on $\\mathbb{R}$ whose one-point motions are weak solutions to a stochastic differential equation $dX(t)=a(X(t))dt+dw(t)$ that move independently before the meeting time and coalesce at the meeting time. We study special modification of such flow (constructed in \\cite{Riabov}) that gives rise to a random dynamical system and thus allows to discuss stationary points. Existence of a unique stationary point is proved in the case of a strictly monotone Lipschitz drift by developing a variant of a pullback procedure. Connections between the existence of a stationary point and properties of a dual flow are discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-17T18:52:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51",
      "60H15",
      "37H15"
    ],
    "keywords": [
      "coalescing stochastic flows",
      "unique stationary point",
      "meeting time",
      "strictly monotone lipschitz drift",
      "stochastic differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}